Damped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation d 2 y d t 2 + 2 λ d y d t + ω 2 y = 0 The zeros of its characteristic equation are m 1 = − λ + λ 2 − ω 2 and m 1 = − λ − λ 2 − ω 2 For λ 2 − ω 2 > 0 , the system is overdamped; λ 2 − ω 2 = 0 , it is critically damped; and for λ 2 − ω 2 < 0 , it is underdamped. Determine whether the differential equation represents an overdamped, critically damped, or underdamped system. Find the particular solution that satisfies the initial conditions. Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system. d 2 y d t 2 + 8 d y d t + 16 y = 0 y ( 0 ) = 1 and y ′ ( 0 ) = 1

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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 11PS
Textbook Problem

Damped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation

d 2 y d t 2 + 2 λ d y d t + ω 2 y = 0

The zeros of its characteristic equation are

m 1 = λ + λ 2 ω 2

and

m 1 = λ λ 2 ω 2

For λ 2 ω 2 > 0 , the system is overdamped; λ 2 ω 2 = 0 , it is critically damped; and for λ 2 ω 2 < 0 , it is underdamped.

Determine whether the differential equation represents an overdamped, critically damped, or underdamped system.

Find the particular solution that satisfies the initial conditions.

Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system.

d 2 y d t 2 + 8 d y d t + 16 y = 0

y ( 0 ) = 1 and y ( 0 ) = 1

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Chapter 16 Solutions

Multivariable Calculus
Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Graphical and Analytic Analysis In Exercises 15...Ch. 16.1 - Graphical and Analytic AnalysisIn Exercises 15 and...Ch. 16.1 - Finding a Particular Solution In Exercises17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Integrating Factor Show that each expression is...Ch. 16.1 - Integrating FactorShow that thedifferential...Ch. 16.1 - Tangent Curves In Exercises 39-42, use agraphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Finding an Equation of a Curve In Exercise 43 and...Ch. 16.1 - Finding an Equation of a Curve In Exercises 43 and...Ch. 16.1 - Cost In a manufacturing process where y=C(x)...Ch. 16.1 - HOW DO YOU SEE? The graph shows several...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - EXPLORING CONCEPTS Exact Differential Equation In...Ch. 16.1 - EXPLORING CONCEPTS Exact Differential Equation In...Ch. 16.1 - Exact Differential Equation Find all nonzero...Ch. 16.1 - Exact Differential Equation Find all nonzero...Ch. 16.1 - Every separable equation is an exact equation.Ch. 16.1 - Every exact equation is a separable equation.Ch. 16.1 - If Mdx+Ndy=0 is exact, then...Ch. 16.1 - If Mdx+Ndy=0 is exact, then xMdx+xNdy=0 is also...Ch. 16.2 - Linear Differential Equation Determine the order...Ch. 16.2 - Linearly Independent Describe what is means for...Ch. 16.2 - Using Zeros The zeros of the characteristics...Ch. 16.2 - Finding a General Solution Explain how to find the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a particular Solution Consider the...Ch. 16.2 - Finding a Particular Solution Determine C and ...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding another Solution Show that the equation...Ch. 16.2 - General Solution of a Differential Equation What...Ch. 16.2 - Several shock absorbers are shown at the right. Do...Ch. 16.2 - Give a geometric argument to explain why the graph...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Real Zeros The characteristics equation of the...Ch. 16.2 - Complex Zero The characteristics equation of the...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.3 - Writing What is the form of the general solution...Ch. 16.3 - Choosing a MethodDetermine whether you woulduse...Ch. 16.3 - Method of Undetermined Coefficients InExercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Choosing the Form of the ParticularSolution In...Ch. 16.3 - Choosing the Form of the ParticularSolution In...Ch. 16.3 - Choosing the Form of the Particular Solution In...Ch. 16.3 - Choosing the Form of the Particular Solution In...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Method of Variation of Parameters InExercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Electrical Circuits In Exercises 29 and 30, use...Ch. 16.3 - Electrical Circuits In Exercises 29 and 30, use...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - EXPLORING CONCEPTS Motion of a Spring Rewrite yh...Ch. 16.3 - HOW DO YOU SEE IT? The figure shows the particular...Ch. 16.3 - Motion of a SpringRefer to the differential...Ch. 16.3 - Solving a Differential Equation Solve the...Ch. 16.3 - True or False? In Exercises 39 and 40, determine...Ch. 16.3 - True or False? In Exercises 39 and 40, determine...Ch. 16.3 - PUTNAM EXAM CHALLENGE For all real x, the...Ch. 16.4 - CONCEPT CHECK Power Series Solution Method...Ch. 16.4 - CONCEPT CHECK Recursion Formula What is a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Finding Terms of a Power Series SolutionIn...Ch. 16.4 - Finding Terms of a Power Series SolutionIn...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - Investigation Consider the differential equation...Ch. 16.4 - HOW DO YOU SEE IT? Consider the differential...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Airys Equation Find the first six terms of the...Ch. 16 - Testing for Exactness In Exercises 1 and 2,...Ch. 16 - Testing for Exactness In Exercises 1 and 2,...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Graphical and Analytical Analysis In Exercise 7...Ch. 16 - Graphical and Analytical Analysis In Exercise 7...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Verifying a Solution In Exercises 17-18, verify...Ch. 16 - Verifying a Solution In Exercises 17 and 18,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Boundary...Ch. 16 - Finding a Particular Solution: Boundary...Ch. 16 - Motion of a SpringIn Exercise 35-36, a 64-pound...Ch. 16 - Motion of a SpringIn Exercise 35-36, a 64-pound...Ch. 16 - Method of Undetermined Coefficients In...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Choosing the Form of the Particular Solution...Ch. 16 - Choosing the Form of the Particular Solution In...Ch. 16 - Method of Undetermined Coefficients In Exercises43...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Electrical Circuit The differential equation...Ch. 16 - Investigation The differential equation...Ch. 16 - Think About It (a) Explain how, by observation,...Ch. 16 - Think About It Explain how you can find a...Ch. 16 - Power Series Solution In Exercises 59 and 60, use...Ch. 16 - Power Series Solution In Exercises 59 and 60, use...Ch. 16 - Approximation by Taylor Series In Exercises 61 and...Ch. 16 - Approximation by Taylor Series In Exercises 61 and...Ch. 16 - Finding a General Solution Find the value of k...Ch. 16 - Using an Integrating Factor The differential...Ch. 16 - Finding a General Solution Find the general...Ch. 16 - Finding a General Solution Find the general...Ch. 16 - Distinct Real Zeros Given that the characteristic...Ch. 16 - Limit of a Solution Given that a and b are...Ch. 16 - Trivial and Nontrivial Solutions Consider the...Ch. 16 - Eulers Differential Equation Eulers differential...Ch. 16 - Pendulum Consider a pendulum of length L that...Ch. 16 - Deflection of a Beam A horizontal beam with a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Airys Equation Consider Airys equation given in...Ch. 16 - ChebyshevsEquation ConsiderChebyshevs equation...Ch. 16 - Bessels Equation: Order Zero The differential...Ch. 16 - Bessels Equation: Order One The differential...Ch. 16 - Hermites Equation Consider Hermites Equation...Ch. 16 - Laguerres Equation Consider Laguerres Equation...

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